Question

defg ~ hjkl. what is the length of \\(\overline{lj}\\)?
two rectangles: defg with de=40 in, dg=27 in; hjkl with lh=18 in
options:
a \\(26\frac{2}{3}\\)
b \\(22\frac{1}{3}\\)
c \\(18\frac{2}{5}\\)
d \\(32\frac{3}{4}\\)

Explanation:

Step1: Set up the proportion

Since \(DEFG\sim HJKL\), the ratios of corresponding sides are equal. So, \(\frac{DE}{HL}=\frac{DG}{LJ}\). Substituting the known values: \(\frac{40}{18}=\frac{27}{LJ}\).

Step2: Cross - multiply

Cross - multiplying gives \(40\times LJ = 18\times27\).

Step3: Solve for \(LJ\)

First, calculate \(18\times27 = 486\). Then, \(LJ=\frac{486}{40}=\frac{243}{20}=12.15\) (This is wrong, let's correct the proportion. The correct proportion is \(\frac{DE}{HJ}=\frac{DG}{LJ}\). Wait, no, for similar rectangles \(DEFG\sim HJKL\), \(DE\) corresponds to \(HL\), \(DG\) corresponds to \(LJ\). Wait no, similar polygons: if \(DEFG\sim HJKL\), then \(\frac{DE}{HL}=\frac{DG}{LJ}\). Wait \(DE = 40\), \(DG=27\), \(HL = 18\). Let's re - do:
The correct proportion is \(\frac{DE}{HL}=\frac{DG}{LJ}\) (corresponding sides of similar polygons). So \(\frac{40}{18}=\frac{27}{LJ}\) (wrong, correct is \(\frac{DE}{HJ}=\frac{DG}{LJ}\). Wait no, for rectangles \(DEFG\) and \(HJKL\), \(DE\) and \(HL\) are one pair of corresponding sides, \(DG\) and \(LJ\) are another pair. Wait no, \(DEFG\sim HJKL\) (assuming the order of letters matters for correspondence). So \(DE\) corresponds to \(HJ\), \(EF\) corresponds to \(JK\), \(FG\) corresponds to \(KL\), \(GD\) corresponds to \(LH\). No, for rectangles (all angles \(90^{\circ}\)), the ratio of length to length and width to width. Let \(DE = 40\) (length of first rectangle), \(DG = 27\) (width of first rectangle), \(HL=18\) (width of second rectangle). Let \(LJ=x\) (length of second rectangle). The proportion is \(\frac{DE}{LJ}=\frac{DG}{HL}\) (length to length and width to width). So \(\frac{40}{x}=\frac{27}{18}\).

Step1: Set up the correct proportion

\(\frac{40}{x}=\frac{27}{18}\) (where \(x = LJ\)).

Step2: Cross - multiply

\(27x=40\times18\).

Step3: Calculate \(40\times18\)

\(40\times18 = 720\). So \(27x = 720\).

Step4: Solve for \(x\)

\(x=\frac{720}{27}=\frac{80}{3}=26\frac{2}{3}\).

Answer:

A. \(26\frac{2}{3}\)